260 
MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OP 
It is evident that | x k I k | tends to zero as \x\ tends to infinity for any finite value 
of Jc. 
In the remaining part of the integral put s + 9 = — y and we find 
G - p (-x;0) = (—y)~^x~ y Y (y+0) dy+I k , 
the integral being taken round a contour C A , which encloses the origin and embraces 
the positive half of the real axis up to the point A. — H (6). 
The poles of T (?/ + #) are at the points — 6—r , r = 0, 1 , 2, ... <x>. 
Hence, within a circle of radius equal to the minimum value of \0 + r\, it admits 
the convergent expansion 
; r {r) (0)y r 
r =0 r! ' 
I will show that (— x ; 6) admits the asymptotic expansion 
,“o27t J r! 
(-yy-Pe-v' 0 ^ dy, 
the integrals being taken round a gamma-function contour which encloses the origin, 
embraces the real axis, and passes from positive infinity to positive infinity again. 
This expansion may evidently be written 
X r=o r ! r (/3—r) (log x) r ~ p+1 
We have, if m be a finite positive integer, 
G /J (— x ; 6) —I k —x 0 % 
( — ) r r <r) (0) 
= — x~ e [ {—y) 
2tt JcH J ' 
,-=o r! r(^-r) (log x ) 
Y (r) (0) r 
r-^ + 1 
~^x~ y r (y + 9) 
. ^ 
r =0 
r ! 
y 
dy— 
^ sin tt/3 
it 
y-p + r x e- V y^i^)dy . ( 1 ), 
the latter integrals l^eing taken along the positive half of the real axis. 
If we denote the sum of these integrals by J, we readily see that, for any finite 
value of m, we can by taking \x\ sufficiently large make \3x k \ as small as we please. 
§ 13. We have now to consider the first integral in (1). 
Let 7 ] be a point on the positive half of the real axis just within the circle of 
convergence of T (y + 6 ), so that | ^ j < the minimum value of | 0 + n \, n = 0, 1, 2,... oo. 
Then the first integral in question can be split up into two others, I x and I 2 (say). 
We denote by l x the integral round a contour M (say), enclosing the origin and 
passing from the point y to this point again, y being on the cross-cut which renders 
the subject of integration uniform. The remaining integral I 2 will be equal to 
sin^r/3 x - 0 
r A-&(0) 
TT 
y »x 
r W _s!» 
r =0 
r\ 
dy. 
